Formalizing the stability of the two Higgs doublet model potential into Lean: identifying an error in the literature

Abstract: In 2006, using the best methods and techniques available at the time, Maniatis, von Manteuffel, Nachtmann and Nagel published a now widely cited paper on the stability of the two Higgs doublet model (2HDM) potential. Twenty years on, it is now easier to apply the process of formalization into an interactive theorem prover to this work thanks to projects like Mathlib and PhysLib (formerly PhysLean and Lean-QuantumInfo), and to ask for a higher standard of mathematical correctness. Doing so has revealed an error in the arguments of this 2006 paper, invalidating their main theorem on the stability of the 2HDM potential. This case is noteworthy because to the best of our knowledge it is the first non-trivial error in a physics paper found through formalization. It was one of the first papers where formalization was attempted, which raises the uncomfortable question of how many physics papers would not pass this higher level of scrutiny.

• The paper’s main contribution is the correction of a widely cited error in the 2HDM stability conditions through a fully mechanized Lean formalization.
• It employs the Gram matrix and field orbit reduction to reformulate the Higgs potential, achieving a significant complexity reduction.
• The approach highlights the critical role of formal proofs in theoretical physics for ensuring absolute mathematical correctness.

Formal Verification of the 2HDM Potential Stability: Correction of an Error via Lean Formalization

Overview

The paper "Formalizing the stability of the two Higgs doublet model potential into Lean: identifying an error in the literature" (2603.08139) initiates a systematic and rigorous formalization of the two Higgs doublet model (2HDM) potential's stability conditions using the Lean interactive theorem prover environment, specifically through the PhysLib library. A principal outcome is the identification of a non-trivial error in a widely cited 2006 work (Maniatis et al.), which purportedly gave necessary and sufficient conditions for the stability of the full 2HDM potential. This work reconstructs the correct mathematical statements and provides a fully formalized complexity reduction of the problem, with major implications for mathematical physics methodologies.

Mathematical Re-Formulation and Formalization Strategy

The methodology replaces the informal calculational approach with fully formalized definitions and proofs in Lean, exploiting type theory-based verification for absolute mathematical correctness. The 2HDM potential is analyzed in terms of orbits in the field space, employing the Gram matrix and the associated 4-vector representation ($K_\mu$), which naturally encapsulates gauge redundancies via the orbit structure under the Standard Model gauge group.

The central mathematical object is the 2HDM scalar potential, given by

$V(\Phi_1, \Phi_2) = V_2 + V_4$

with explicit parameterization in gauge-invariant terms, and subsequently re-expressed as a quadratic form in $K_\mu$:

$$V(K) = \sum_{\mu} \xi_\mu K_\mu + \sum_{\mu,\nu} \eta_{\mu\nu} K_\mu K_\nu$$

with all parameters mapped from the original Lagrangian coefficients to the new basis. The formalization extends to showing surjectivity of the Gram vector mapping and classifying field configurations up to gauge orbits.

Correct Stability Conditions and Complexity Reduction

Stability is formalized as the existence of a real lower bound $c$ such that for all Higgs field configurations,

$V(\Phi_1, \Phi_2) \geq c.$

Through formal reduction, this is re-cast in terms of $K_\mu$, then further reduced by normalization to a vector $\vec{k}$ with norm $\leq 1$ and an overall positive $K_0$ scaling:

$$c \leq K_0 J_2(\vec{k}) + K_0^2 J_4(\vec{k}),\quad \lVert \vec{k} \rVert^2 \leq 1,\ K_0 > 0.$$

The authors supply a new, completely verified complexity reduction (not previously in the literature): the full potential is stable if and only if there exists $c\geq0$ such that for all $\lVert \vec{k} \rVert\leq1$,

• $J_4(\vec{k}) \geq 0$
• If $J_2(\vec{k}) < 0$, then $J_2^2(\vec{k}) \leq 4cJ_4(\vec{k})$

This reduction minimizes the number of quantifiers, providing an optimal representation for subsequent computational or analytic work.

Critical Examination of Prior Literature

The core contribution is the rigorous demonstration that the sufficiency claim in Theorem 1 of Maniatis et al. (2006) is false. Specifically, their "condition C":

• For all $\vec{k}$ with $\lVert\vec{k}\rVert\leq 1$, either $J_4(\vec{k}) > 0$ or [$J_4(\vec{k}) = 0$ and $J_2(\vec{k})\geq 0$]

is only necessary, not sufficient, for absolute stability. The authors construct an explicit counterexample: a selection of mass and quartic parameters yielding a potential that satisfies condition C throughout the relevant domain but is, in fact, unbounded from below. The instability is certified by explicit test configurations and fully mechanized Lean proofs.

Implications and Theoretical Significance

Impact on Theoretical and Mathematical Physics

• Formal Proof over Heuristic Reasoning: This study exemplifies the critical value of ITPs in physics, highlighting that reasoning chains in even highly cited, mathematically careful papers are not immune to subtle but material errors.
• Gold Standard for Mathematical Rigor: The results encourage a paradigm shift toward formal proof tools when assessing non-trivial mathematical conditions in theoretical physics, especially for questions of stability and positivity which underpin model viability in high-energy theory.
• Complexity Characterization: The formal quantifier hierarchy analysis clarifies not only the logical content but also the computational nature of the stability decision problem. The optimal complexity reduction aids both analytical approaches and prospective computer algebra verifications for extended Higgs sectors or generalizations.

Future Directions

• Extending to Broader Model Spaces: The methodology is directly extensible to cases with additional Higgs fields, singlet extensions, or more elaborate symmetry-breaking sectors.
• Automated Physics Proofs: While this work is primarily human-verified, future developments may increasingly leverage AI-assisted proof search, provided such tools yield transparency and verifiability.
• Community Libraries: The open sourcing of all definitions, lemmas, and proofs in PhysLib sets a blueprint for community-driven, peer-verifiable physics theorem repositories.

Conclusion

This work refines the logical foundations for analyzing the boundedness of the 2HDM potential, supplanting a widely referenced but incorrect sufficiency criterion with a formally verified necessary and sufficient condition. The correction has direct pedagogical and practical value for studies in extended Higgs sector phenomenology. More fundamentally, the paper underscores the nontrivial risk of unformalized results, motivating the integration of ITP-based proof standards into mainstream mathematical physics research (2603.08139).